Finite Group Databasebeta

← #10830409

Subgroups — #10830409

Order 768.

1366 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85818C211yesmaximal, characteristic
2C3×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3C2×#5095C212yesmaximal, complemented
4#82441C212yesmaximal, complemented
5#71637C211yesmaximal, characteristic, complemented
6#69694C212yesmaximal, complemented
7#69692C212yesmaximal, complemented
8#69593C212yesmaximal, complemented
9#69395C212yesmaximal, complemented
10#24225—31nomaximal, Sylow
11C2×#1007C2212yes—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4×Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×#942C2214yes—
16#5577—22no—
17#5576—24no—
18Q8×D12—22no—
19#5510—22no—
20#5510C2212yes—
21#5477—22no—
22#5477C2212yes—
23C3×#391C2211yescharacteristic
24C3×#391C2211yescharacteristic
25C3×#389—22no—
26C3×#384C2211yescharacteristic
27C3×C4×C4⋊C4—22no—
28C3×C43C2211yescharacteristic
29C2×#836C2212yes—
30#5095—28no—
31#5031—22no—
32#5028—24no—
33#5027—22no—
34#4966C2214yes—
35C2×#805C2214yes—
36C2×#805C2214yes—
37#4924C2212yes—
38#4530C2212yes—
39#3497—31no—
40#3037—32no—
41C2×#456—32no—
42C4×C4⋊Q8D311yescharacteristic, complemented
43#2039—32no—
44#2007—32no—
45#1962—32no—
46#1801—31no—
47#1793—32no—
48C22×C3×Q8C2311yescharacteristic
49C22×C3×Q8C2311yescharacteristic
50#1007—24no—
51#1007—24no—
52#1007—44no—
53C2×#239—24no—
54C2×D3×Q8—24no—
55C2×#235—24no—
56C2×#235C2314yes—
57C3×C4⋊Q8—22no—
58C3×C4⋊Q8D414yescomplemented
59C3×C4×Q8—22no—
60C3×C4×Q8—22no—
61C3×C4×Q8—22no—
62C3×C4×Q8—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4D412yes—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C42—21no—
70C2×C3×C42—21no—
71C2×C3×C42—21no—
72C2×C3×C42D412yes—
73C2×C3×C42C2311yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×#216—24no—
77#942—28no—
78#942—44no—
79C2×#215—24no—
80#930—22no—
81#930—24no—
82#930—22no—
83C2×#208C2314yes—
84#897—28no—
85#897—24no—
86#895—28no—
87#895—24no—
88#875—24no—
89#875D414yes—
90#874—22no—
91#874D412yes—
92C4×D12—24no—
93C4×D12—24no—
94C4×D12—22no—
95C4×D12—22no—
96#869—22no—
97#869D412yes—
98C3×#95—22no—
99C3×#95—22no—
100#836—24no—
101#810—24no—
102#809—24no—
103#805—28no—
104#805—44no—
105C4⋊(C3⋊C8)—22no—
106C4⋊(C3⋊C8)—22no—
107C2×2−1+4—32no—
108C2×#137—34no—
109#550—62no—
110D4×Q8—62no—
111#548—64no—
112#537—32no—
113#537—62no—
114#532—32no—
115#532—62no—
116C2×C4⋊Q8C2×D311yescharacteristic
117C2×C4×Q8C2×D311yescharacteristic
118C2×C4×Q8C2×D311yescharacteristic
119#464—64no—
120#462—62no—
121#460—62no—
122#456—68no—
123#423—32no—
124C2×#104—34no—
125C2×#104—34no—
126#417—34no—
127C2×#101—32no—
128#391C2×D311yescharacteristic
129#391C2×D311yescharacteristic
130#389—22no—
131#384C2×D311yescharacteristic
132C4×C4⋊C4—22no—
133C43C2×D311yescharacteristic
134#343—32no—
135C2×C3×Q8—22no—
136C2×C3×Q8—21no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8C2×D412yes—
141C2×C3×Q8C2×D412yes—
142C2×C3×Q8#4112yes—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D412yes—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2411yescharacteristic
152C2×#67—24no—
153#239—48no—
154#239—48no—
155D3×Q8—48no—
156D3×Q8—48no—
157#235—24no—
158#235—24no—
159#235—24no—
160#235—24no—
161#235—48no—
162#235—48no—
163C2×D12—24no—
164C2×D12—22no—
165C2×D12—24no—
166C2×D12—22no—
167C2×D12C2×D414yes—
168C2×C4×D3—28no—
169C2×C4×D3—24no—
170C2×Dic6—24no—
171C2×Dic6C2×D414yes—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—28no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—44no—
177C3×C4⋊C4—42no—
178C3×C42—24no—
179C3×C42—22no—
180C3×C42—22no—
181C3×C42—21no—
182C3×C42—22no—
183C3×C42—42no—
184C3×C42—41no—
185C3×C42—41no—
186C3×C42C2×D412yes—
187C3×C42C2×D412yes—
188C3×C42C2×D412yes—
189#216—48no—
190#215—48no—
191(C2×D3)⋊C4—28no—
192(C2×D3)⋊C4—24no—
193(C2×D3)⋊C4—24no—
194C4⋊Dic3—24no—
195C4⋊Dic3—22no—
196C4⋊Dic3—22no—
197#208—24no—
198#208—24no—
199C2×C3⋊C8—24no—
2002−1+4—64no—
2012−1+4—64no—
2022−1+4—124no—
203C2×#41—34no—
204C2×#41—64no—
205C2×#41—64no—
206C22×Q8—64no—
207C22×Q8C22×D311yescharacteristic
208C22×Q8C22×D311yescharacteristic
209#137—68no—
210#137—124no—
211C2×Q16—64no—
212C2×SD16—64no—
213C2×M16—34no—
214C4⋊Q8—22no—
215C4⋊Q8—32no—
216C4⋊Q8—62no—
217C4⋊Q8D1214yescomplemented
218#127—32no—
219#127—62no—
220#124—34no—
221#124—64no—
222C22⋊Q8—64no—
223C22⋊Q8—68no—
224#121—64no—
225#121—68no—
226#121—62no—
227#121—62no—
228#121—64no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×D4—64no—
234C4×D4—64no—
235C4×D4—62no—
236C4×D4—62no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4#6712yes—
241C2×C4⋊C4C22×D311yescharacteristic
242C2×C4⋊C4C22×D311yescharacteristic
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42#6712yes—
247C2×C42C22×D311yescharacteristic
248C2×C42C22×D311yescharacteristic
249C2×C42C22×D311yescharacteristic
250C4⋊C8—62no—
251C4⋊C8—62no—
252#104—68no—
253#104—124no—
254#103—64no—
255#102—64no—
256#101—64no—
257#95—22no—
258#95—22no—
259C23×C3C22×D411yescharacteristic
260C22×D3—24no—
261C3×Q8—24no—
262C3×Q8—24no—
263C3×Q8—24no—
264C3×Q8—42no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4#12012yes—
283C2×C3×C4#12011yescharacteristic
284C2×C3×C4#12011yescharacteristic
285C2×C3×C4#12114yes—
286C2×C3×C4C22×D411yescharacteristic
287C2×C3×C4C22×D412yes—
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C4H(5,𝔽2)12yes—
290#67—44no—
291#67—44no—
292C2×Dic3—24no—
293D12—24no—
294D12—24no—
295D12—48no—
296D12—44no—
297D12—82no—
298D12—84no—
299C4×D3—416no—
300C4×D3—44no—
301C4×D3—44no—
302Dic6—24no—
303Dic6—24no—
304Dic6—48no—
305C3⋊C8—44no—
306#41—64no—
307#41—64no—
308#41—64no—
309#41—64no—
310#41—128no—
311#41—128no—
312#41—128no—
313#41—128no—
314C2×Q8—21no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—34no—
320C2×Q8—64no—
321C2×Q8—128no—
322C2×Q8—128no—
323C2×Q8C2×D1212yes—
324C2×Q8C2×D1212yes—
325C2×Q8#23512yes—
326C2×D4—34no—
327C2×D4—64no—
328C2×D4—64no—
329C2×D4—64no—
330C2×D4—62no—
331C2×D4—62no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—68no—
337C22×C4—64no—
338C22×C4C2×#6712yes—
339C22×C4C2×#6711yescharacteristic
340C22×C4C2×#6712yes—
341C22×C4C2×#6711yescharacteristic
342C22×C4C23×D311yescharacteristic
343Q16—128no—
344SD16—128no—
345M16—64no—
346M16—64no—
347C2×C8—64no—
348C4⋊C4—28no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—44no—
353C4⋊C4—42no—
354C4⋊C4—64no—
355C4⋊C4—62no—
356C4⋊C4—62no—
357C22⋊C4—68no—
358C22⋊C4—64no—
359C22⋊C4—64no—
360C42—24no—
361C42—22no—
362C42—22no—
363C42—21no—
364C42—22no—
365C42—42no—
366C42—41no—
367C42—41no—
368C42C2×D1212yes—
369C42D3×D412yes—
370C42D3×D412yes—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3C2×#12011yescharacteristic
374C22×C3C2×#12111yescharacteristic
375C22×C3#53511yescharacteristic
376C2×D3—44no—
377C2×D3—44no—
378C2×D3—44no—
379C3×C4—24no—
380C3×C4—24no—
381C3×C4—21no—
382C3×C4—22no—
383C3×C4—42no—
384C3×C4—42no—
385C3×C4—42no—
386C3×C4—81no—
387C3×C4#45612yescomplemented
388C3×C4D4212yes—
389C3×C4#54612yes—
390Dic3—44no—
391C23—64no—
392C23C22×#6711yescharacteristic
393Q8—24no—
394Q8—24no—
395Q8—24no—
396Q8—42no—
397Q8—64no—
398Q8—64no—
399Q8—128no—
400D4—64no—
401D4—64no—
402D4—128no—
403D4—124no—
404D4—124no—
405D4—124no—
406D4—242no—
407D4—244no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—64no—
426C2×C4—1216no—
427C2×C4—124no—
428C2×C4—124no—
429C2×C4#93014yes—
430C2×C4#93711yescharacteristic
431C2×C4#93712yes—
432C2×C4#93711yescharacteristic
433C2×C4C22×D1211yescharacteristic
434C2×C4C2×#23511yescharacteristic
435C2×C4C2×D3×D412yes—
436C2×C4#100912yes—
437C8—124no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3#240811yescharacteristic
441C2×C3C2×#45611yescharacteristic
442C2×C3#301811yescharacteristic
443D3—84no—
444C22—21no—
445C22—21no—
446C22—124no—
447C22—124no—
448C22—124no—
449C22C2×#93011yescharacteristic
450C22C2×#93711yescharacteristic
451C22#583011yescharacteristic
452C4—24no—
453C4—21no—
454C4—22no—
455C4—24no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—81no—
460C4—124no—
461C4#509512yescomplemented
462C4D4×D1212yes—
463C4#555012yes—
464C3#2422511yescharacteristic, complemented, Sylow
465C2—21no—
466C2—21no—
467C2—244no—
468C2#8240311yescharacteristic
469C2C2×#509511yescharacteristic
470C2#8332211yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×C4⋊Q8C2
socleC22×C3C2×#120
prime radical#85818C2
hypercenterC4×C4⋊Q8D3
hypocenterC3#24225
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×C4⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×C4×C4⋊Q8C2semidirect
3. C2×#5095C2semidirect
4. #82441C2semidirect
5. #71637C2semidirect
6. #69694C2semidirect
7. #69692C2semidirect
8. #69593C2semidirect
9. #69395C2semidirect
42. C4×C4⋊Q8D3semidirect
58. C3×C4⋊Q8D4semidirect
217. C4⋊Q8D12semidirect
387. C3×C4#456semidirect
461. C4#5095semidirect
464. C3#24225semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#858182
C3×C4×C4⋊Q82
C2×#50952
#824412
#716372
#696942
#696922
#695932
#693952
#242253